$\lim_{k\to\infty} X^{(k)}=A^{-1}$ if and only if $ρ(I − AX^{(0)}) < 1$

Question

A is $n × n$ nonsingular matrix, and $X^{(0)}$ is n × n matrices and sequence is defined

$X^{(k+1)} = X^{(k)} + X^{(k)}(I − AX^{(k)}),$ $k=0,1,2.....$

I want to show that $\lim_{k\to\infty} X^{(k)}=A^{-1}$ if and only if $ρ(I − AX^{(0)}) < 1$ where $ρ$ is spectral radius.

I'm stuck on the problem. How can I solve it?

Answer 1

Hints. Prove by mathematical induction that $I-AX^{(k)}=(I-AX^{(0)})^{2^k}$. Now the RHS tends to zero as $k\to\infty$, because $\rho(I-AX^{(0)})<1$.